The denominator of a rational number is greater than its numerator by 4. If 4 is subtracted from the numerator and 2 is added to its denominator, the new number becomes `1/6`. Find the original number.

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information provided.

### Step 1: Define the Variables
Let the numerator of the rational number be \( N \) and the denominator be \( D \).

### Step 2: Set Up the First Equation
According to the problem, the denominator is greater than the numerator by 4. This can be expressed as:
\[ D = N + 4 \]

### Step 3: Set Up the Second Equation
The problem states that if 4 is subtracted from the numerator and 2 is added to the denominator, the new fraction becomes \( \frac{1}{6} \). This can be expressed as:
\[ \frac{N - 4}{D + 2} = \frac{1}{6} \]

### Step 4: Substitute the First Equation into the Second Equation
We can substitute \( D \) from the first equation into the second equation:
\[ \frac{N - 4}{(N + 4) + 2} = \frac{1}{6} \]
This simplifies to:
\[ \frac{N - 4}{N + 6} = \frac{1}{6} \]

### Step 5: Cross-Multiply
Cross-multiplying gives us:
\[ 6(N - 4) = 1(N + 6) \]
This expands to:
\[ 6N - 24 = N + 6 \]

### Step 6: Rearrange the Equation
Now, we will rearrange the equation to isolate \( N \):
\[ 6N - N = 24 + 6 \]
This simplifies to:
\[ 5N = 30 \]

### Step 7: Solve for \( N \)
Now, divide both sides by 5:
\[ N = 6 \]

### Step 8: Find \( D \)
Now that we have \( N \), we can find \( D \) using the first equation:
\[ D = N + 4 = 6 + 4 = 10 \]

### Step 9: State the Original Number
The original rational number is:
\[ \frac{N}{D} = \frac{6}{10} = \frac{3}{5} \]

### Final Answer
The original number is \( \frac{3}{5} \).

---